# Help with Summation: Evaluate 1/4+2/16+3/64+4/256+5/1024+...

• darkvalentine
In summary, the given summation can be written as Sum(k=1 to infinity, k/(4^k)). By using the formula for summing a geometric series, it can be simplified to a function f(x). The derivative of this function, f'(x), evaluated at x=1 can then be used to calculate the given summation.
darkvalentine

## Homework Statement

Evaluate: 1/4+2/16+3/64+4/256+5/1024+...

## The Attempt at a Solution

The summation can be written as: Sum(k=1 to infinity, k/(4^k))

You know how to sum x^k/(4^k), right? It's a geometric series. It gives you some function f(x). Now consider what f'(x) is evaluated at x=1.

Dick said:
You know how to sum x^k/(4^k), right? It's a geometric series. It gives you some function f(x). Now consider what f'(x) is evaluated at x=1.

Honestly I do not know how to sum x^k/(4^k), can you explain a little more why we have to put it in a function f(x)? f'(x) at x=1 going to be (k-ln4)/(4^k) but then ?

darkvalentine said:
Honestly I do not know how to sum x^k/(4^k), can you explain a little more why we have to put it in a function f(x)? f'(x) at x=1 going to be (k-ln4)/(4^k) but then ?

The sum of x^k/4^k is geometric because it's the sum of (x/4)^k. Look up the formula for summing a geometric series. The common ratio r=x/4, yes? The result is a function of r, which x/4. So it's a function of x. And when I say f'(x) I mean the derivative with respect to x. Isn't it sum k*x^(k-1)/4^k? No logs needed. Do you see how the k in your sum comes in?

Dick said:
The sum of x^k/4^k is geometric because it's the sum of (x/4)^k. Look up the formula for summing a geometric series. The common ratio r=x/4, yes? The result is a function of r, which x/4. So it's a function of x. And when I say f'(x) I mean the derivative with respect to x. Isn't it sum k*x^(k-1)/4^k? No logs needed. Do you see how the k in your sum comes in?

Thanks, I got it ^^

## What is summation and how is it used in science?

Summation, also known as sigma notation, is a mathematical process used to add up a series of numbers. It is commonly used in science to calculate the total value of a variable over a certain range or to find the average of a set of data.

## What is the formula for summation?

The formula for summation is Σ (upper limit, lower limit) f(x), where f(x) is the function being summed and the upper and lower limits represent the range of values to be added.

## How do I evaluate a summation?

To evaluate a summation, you can use the formula Σ (upper limit, lower limit) f(x) and replace f(x) with the given function. In this case, the function is 1/4, 2/16, 3/64, etc. You can also use a calculator or a spreadsheet to perform the calculations.

## What is the purpose of evaluating this specific summation: 1/4+2/16+3/64+4/256+5/1024+...?

This specific summation is known as a geometric series and it is often used in science to model real-world phenomena such as population growth or radioactive decay. Evaluating this summation can help us understand and predict the behavior of these systems.

## What are some real-world applications of summation in science?

Summation is used in a variety of scientific fields, including physics, chemistry, biology, and economics. Some examples of its applications include calculating the total force on an object, finding the average temperature over a period of time, and determining the total cost of a chemical reaction. It is also used in data analysis to find patterns and trends in large sets of data.

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